Two arcs of two different circle are of equal lengths. If these arcs subtends angles of `45^(@) and 60^(@)` at the centre of the circles. Find the ratio of the radii of the two circles.

Answer

To solve the problem step by step, we will use the relationship between the arc length, radius, and the angle subtended at the center of the circle.

### Step 1: Understand the problem
We have two arcs of different circles that are of equal lengths. The angles subtended by these arcs at the centers of their respective circles are given as \(45^\circ\) and \(60^\circ\).

### Step 2: Convert angles from degrees to radians
To use the formula for arc length, we need to convert the angles from degrees to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
- For \(45^\circ\):
\[ \theta_1 = 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4} \]
- For \(60^\circ\):
\[ \theta_2 = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \]

### Step 3: Write the formula for arc length
The arc length \(L\) of a circle can be calculated using the formula:
\[ L = r \times \theta \]
where \(r\) is the radius and \(\theta\) is the angle in radians.

### Step 4: Set up the equations for arc lengths
Let \(L_1\) be the arc length of the first circle and \(L_2\) be the arc length of the second circle. Since both arcs are of equal length, we can write:
\[ L_1 = r_1 \times \theta_1 \]
\[ L_2 = r_2 \times \theta_2 \]
Given that \(L_1 = L_2\), we have:
\[ r_1 \times \theta_1 = r_2 \times \theta_2 \]

### Step 5: Rearrange to find the ratio of the radii
We can rearrange the equation to find the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{\theta_2}{\theta_1} \]

### Step 6: Substitute the values of \(\theta_1\) and \(\theta_2\)
Now, substituting the values we found:
\[ \frac{r_1}{r_2} = \frac{\frac{\pi}{3}}{\frac{\pi}{4}} = \frac{\frac{1}{3}}{\frac{1}{4}} = \frac{4}{3} \]

### Step 7: Conclusion
The ratio of the radii \(r_1\) to \(r_2\) is:
\[ r_1 : r_2 = 4 : 3 \]